Embedded in this last statement is the central assumption of this reconstruction, which is the adoption of a Hamiltonian framework for evolving \(\phi \)  — or more technically, the introduction of a symplectic structure on the joint space of \(\rho \)  and \(\phi \) . An advantage of this approach is that it allows us to borrow from schemes developed for classical physics in pursuit of a manifestly relativistic ED that is both statistical and quantum. Of particular relevance to us are the covariant Hamiltonian methods pioneered by Dirac [1]}[2]}, Kuchař [3]}, Teitelboim [4]}[5]}, and others [6]}. These approaches are, in short, characterized by expressing the local Lorentz invariance of space-time in terms of a new symmetry — foliation invariance. In turn, this requirement is implemented by a consistency condition called path independence: the evolution of an initial to a final state is independent of the intervening choice of foliation. The result is a collection of Poisson brackets and constraints that are to be satisfied by the aforementioned local Hamiltonian generators.
